Re: On proving a set is countable

The "question", I thought, was quite clearly stated:
"Could we say that, consequently, a set A is countable if there is a function which establishes a one-to-one correspondence between A
and another countable set?" and the answer to that is "yes". The rest was an example: having shown a bijection between $\displaystyle S_n$ and $\displaystyle Z$, it follows that, since Z is countable, that $\displaystyle S_n$ is countable.

(Perhaps it is the fact that m/n depends upon both m and n that bothers you? That bothered me at first until I realized that the question was about $\displaystyle S_n$ in which n is fixed, and its members depend only on m.)